Question: Solve for $x$ : $ 2|x + 8| - 9 = -4|x + 8| + 3 $
Add $ {4|x + 8|} $ to both sides: $ \begin{eqnarray} 2|x + 8| - 9 &=& -4|x + 8| + 3 \\ \\ { + 4|x + 8|} && { + 4|x + 8|} \\ \\ 6|x + 8| - 9 &=& 3 \end{eqnarray} $ Add ${9}$ to both sides: $ \begin{eqnarray} 6|x + 8| - 9 &=& 3 \\ \\ { + 9} &=& { + 9} \\ \\ 6|x + 8| &=& 12 \end{eqnarray} $ Divide both sides by ${6}$ $ \dfrac{6|x + 8|} {{6}} = \dfrac{12} {{6}} $ Simplify: $ |x + 8| = 2$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 8 = -2 $ or $ x + 8 = 2 $ Solve for the solution where $x + 8$ is negative: $ x + 8 = -2 $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& -2 \\ \\ {- 8} && {- 8} \\ \\ x &=& -2 - 8 \end{eqnarray} $ $ x = -10 $ Then calculate the solution where $x + 8$ is positive: $ x + 8 = 2 $ Subtract ${8}$ from both sides: $ \begin{eqnarray} x + 8 &=& 2 \\ \\ {- 8} && {- 8} \\ \\ x &=& 2 - 8 \end{eqnarray} $ $ x = -6 $ Thus, the correct answer is $x = -10 $ or $x = -6 $.